An explicit trace formula of Jacquet-Zagier type for Hilbert modular forms
Abstract.
We give an exact formula of the average of adjoint -functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier’s formula known for the case of elliptic cusp forms on . As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint -functions.
Key words and phrases:
trace formulas, adjoint -functions.2010 Mathematics Subject Classification:
Primary 11F72; Secondary 11F67.1. Introduction
1.1. Background and motivation
In [35], Zagier proposed an elegant way to compute the traces of Hecke operators on the space of elliptic cusp forms by means of the Rankin-Selberg method. This is a more direct way than the Selberg’s one because it does not need a deliberate rearrangement for renormalization of divergent terms which are inevitably produced by truncation process. Later, a similar study was conducted for Maass forms on in [36], and for general cusp forms on the adelization with an arbitrary number field in the work of Jacquet and Zagier [12]. The formula proved in [12] can be viewed as an “arithmetic deformation” by a complex parameter of the usual Arthur-Selberg trace formula for because the latter one is expected to be recovered from the former one as the residue at . Although their general formula is less explicit than Zagier’s one, it provides us with a different proof of holomorphicity of the symmetric square -function of a cuspidal representation of , which was first proved by Shimura [22] in a classical setting and later was generalized to an adelic setting by [6]. The paper [35] also gave an application of the formula to the proof of the algebraicity of the critical values of the symmetric square -functions for elliptic cusp forms, which was independently obtained by Strum [23] based on [22]. Mizumoto [18] and Takase [29] extended Zagier’s method to Hilbert cusp forms under the assumption that the narrow class number of the base field is one. For application to special values, the explicit nature of their formula is crucial.
In this paper, motivated by these works and intending further potential applications, we shall calculate Jacquet-Zagier’s trace formula for holomorphic Hilbert cusp forms as explicitly as possible when the level is square-free without the assumption on the class number of the base field . For a technical reason, we assume that the prime splits completely in . We remark that this assumption is mild enough to include the interesting cases and with and , and our formulas for Hilbert modular forms with large levels are new even for . Since we use the matrix coefficients of discrete series representations at archimedean places, which are not compactly supported contrary to the test functions dealt in [12], we have to modify the convergence proof of the geometric side in a substantial way. Moreover, we completely calculate all local terms in the formula for a large class of test functions. As an application, we prove an equidistribution result of Satake parameters in the ensemble defined by the symmetric square -functions of holomorphic Hilbert cusp forms of a fixed weight with the varying square-free levels, in such a way that a part of the famous Serre’s equidistribution theorem ([19, Théorème 1]) is recovered from our formula by the specialization at . The non-vanishing of the symmetric square -function of an elliptic modular form at a point in the critical strip has been pursued by many authors ([13], [15], [2]). As a corollary to our asymptotic formula, we obtain infinitely many Hilbert cusp forms with a fixed weight of growing levels whose symmetric square -functions are non-vanishing at a given point in the critical strip. Our method is not the Rankin-Selberg method in the accurate sence, because the Eisenstein series is not unfolded on the convergence region as was done in [35] and [12]; actually, the same proof works if the Eisenstein series is replaced with a Maass cusp form. To illustrate the robustness of our method, we deduce Theorem 9.1 which is viewed as an adelic version of Gon’s formula in an opposite setting to [8].
1.2. Description of main results
Let us state our main result introducing notation used in this article on the way. Let be the set of all positive integers and we write for . For any condition , we put if is true, and if is false, respectively. We set and . All the fractional ideals appearing in this paper are supposed to be invertible.
Let be a totally real number field, the integer ring of and the adele ring of . The completed Dedekind zeta function of is denoted by . (All -functions in this article are supposed to be completed by appropriate gamma factors.) The set of finite places and the set of archimedean places of are denoted by and , respectively. Set . The completion of at is denoted by and the modulus of is by . Then, defines the idele norm on . When , , and denote the maximal order of , the maximal ideal of and , respectively. For a non-zero ideal , let denote the absolute norm of and the set of dividing . Let be the absolute discriminant of and the character of defined as where is the character of such that for . Let , and be -subgroups of defined symbolically as , and ; is a Borel subgroup of . Let be the center of . For , let denote if and if . Set viewed as a subgroup of the adelization . Given a non-zero ideal and an even weight , let be the set of all those irreducible cuspidal automorphic representations of such that is a discrete series representation of of weight for all and has a non-zero vector invariant by the group for all . Let for a while. Set . For a -spherical irreducible representation of with trivial central character, we define
| (1.1) |
with the Satake parameter of ; note that the exponent is determined only up to sign. Let denote the normalized induced representation . Then and if is generic and unitarizable. For , let denote the real-valued character of corresponding to the extension by local class field theory; it depends only on the coset . For and (such that ), we define complex-valued functions () and on as
and
| (1.2) | ||||
We remark that, when viewed as meromorphic functions in , the singularities of these functions at are removable. A computation reveals
| (1.3) |
For a finite subset , a square-free ideal such that , an element , and a non-zero fractional ideal , the product
is well-defined due to (1.3), where is the idele corresponding to and . Let . We define a complex-valued function on as
where is the Legendre function of the 1st kind which is defined for points outside the interval of the real axis ([16, §4.1]).
Suppose is square-free from now on. For and a finite set disjoint from , set , an element of . We set
where denotes the conductor of . Since by the unitarity of , we have when is a non-negative real number. Let be the symmetric square -function of , which on is defined by the Euler product of over completed by the gamma factor over and by finitely many factors over ; it is known to be entire on and is identified with the standard -function of an automorphic representation of ([6]). We have the functional equation . In particular, the sign of the functional equation is plus. Let be a meromorphic function on defined as
| (1.4) | ||||
where
| (1.5) |
For , let be the completed -function of the class field character of the quadratic extension , the relative discriminant of , and a fractional ideal uniquely determined by .
Theorem 1.1.
Let with for all and a non-zero square-free ideal of . Let be a finite subset of such that . We suppose splits completely in and for all . Then, for and such that and , we have the identity
where the terms on the right-hand side are described as follows. The first term is defined as
The second is given by the absolutely convergent sum
where is the totally positive units of the -integer ring of . The third term is given by the absolutely convergent sum
where runs over different cosets such that , for all , and for all with being unramified and non-trivial.
As a corollary of Theorem 1.1, we have a generalization of [35], [18] and [29]. Let be the space of holomorphic functions on such that . Let denote the holomorphic -form on , and let be the contour . For , set
For a finite subset , let be a pure tensor of , viewed as a function on . For a square-free ideal such that , an element , and a non-zero fractional ideal and , set
and
| (1.6) |
Corollary 1.2.
Let , , be as in Theorem 1.1. We suppose splits completely in and for all . Then for and such that , we have the identity
where is the number defined as (1.5) and the terms on the right-hand side are described as follows. The first term is defined as
The second is given by the absolutely convergent sum
The third term is given by the absolutely convergent sum
where and runs over the same set as in Theorem 1.1.
Corollary 1.2 includes known formulas as a special case. Indeed, it is confirmed that [35, Theorem 1], [18, (4.6) and (5.9)], and [29, Proposition 2] are all recovered from Corollary 1.2. Moreover, we can deduce an explicit trace formula of Hecke operators in a more general setting than in these works.
1.2.1. A limit formula
For a real number and a square-free ideal , let us define a discrete Radon measure on as
where , and . Note that . The value is non-negative for lying in , the closure of the absolute convergence region of the Euler product. In some cases, it is more enlightening to work with the assumption
(P) : for any and for all holomorphic cuspidal automorphic representations of with weight ,
which ensures that the measure is non-negative for . The statement (P) is not proved up to now, however it is highly expected to be true from the entireness of ([22], [6]) combined with the Riemann hypothesis of the -function .
For and , we define a non-negative Radon measure on as
Note that the measure coincides with the limit measure in Serre’s theorem [19, Théorème 1]. For , set
and if and . Let be the space of complex-valued polynomial functions on ; by the Stone-Weierstrass theorem, is a dense subspace of . Fix a set of prime ideals of different from and mapped bijectively to the ideal class group of ; this is possible by Chebotarev’s density theorem.
Theorem 1.3.
Let be as in Corollary 1.2. Let be such that . Let be a finite set of non-dyadic finite places of . Let be a square-free ideal relatively prime to the ideals , and .
-
(1)
Let . As ,
Under the condition (P), the same limit formulas are true for all , i.e., the measure converges -weakly to on .
-
(2)
Assume (P). Let be a family of closed subintervals of and . There exists such that for any prime ideal with and relatively prime to and for any , there exists with the following properties: , , and . When , the same conclusion holds without (P).
1.3. Organization of paper
In §2, we construct a kernel function on which represents the resolvent of the product of the shifted normalized Hecke operators at depending on a set of parameters and acting on the space of Hilbert modular cusp forms on of a fixed weight invariant by the open compact subgroup . In §3, we introduce the smoothed Eisenstein series on depending on an entire function vertically of rapid decay. Our definition is similar to but a bit different from the usual definition of the wave-packet in that we start from the normalized Eisenstein series. By a constraint on posed to cancel the poles of the Eisenstein series, our is shown to be rapidly decreasing on a Siegel domain (Proposition 3.2). As in [12], we compute the inner-product of and the diagonal restriction of on in two ways obtaining its two expressions referred to the spectral side and the geometric side. The spectral side is constructed in §3 without any difficulty by the Rankin-Selberg integral; the main issue here is the calculation of local zeta integrals for old forms. The computation of the geometric side is harder to accomplish. Since our test function is not compactly supported, the argument in [12] to deal with the unipotent and the hyperbolic terms by the Poisson summation formula is not applied as it is. By a counter shifting argument inspired by a similar analysis in [21], we entirely circumvent the usage of the Poisson summation formula. The computations of unipotent terms and hyperbolic terms are done in §5 and §6, respectively: We prove the absolute convergence of the hyperbolic term by estimating local orbital integrals whose exact formulas are calculated in §10. By the local multiplicity one theorem of the Waldspurger model ([32, Proposition 9’], [33], [34], [4, §1]), the computation of the elliptic term boils down to the determination of the Eisenstein periods along elliptic tori and the calculation of local orbital integrals. In §7, we recall the formula of Eisenstein period originally due to Hecke, providing a proof in a modern style, and establish the absolute convergence of the elliptic term, granting the formulas of local orbital integrals to be proved in §10. The main results are proved in §8. In §9, we state Theorem 9.1 which should be regarded as a version of Gon’s formula. Since its proof is almost identical to Corollary 1.2 and is much easier than that in some aspect, only a brief indication for proof will be given there. We suggest the readers to move to § 9 to see the statement of Theorem 9.1 after finishing this introduction. Theorem 9.1 has an application to non-vanishing -values, which we refer to our forthcoming work [28].
1.4. Notation
Throughout this paper, we adapt Vinogradov’s notation . For any complex-valued functions and on a set , we write if there exists a constant independent of such that for all . We write when both and hold. If we emphasize dependence of the implied constant on some parameters , we write .
2. Construction of the kernel function
The Haar measures we work with in this article are fixed in the following way. For , let be the additive Haar measure of such that if and if , where is the exponent of the local different of . Fix measures on and on by and by , respectively. We fix Haar measures on and accordingly. We endow , , and with measures , , and , respectively by setting if , if , and by requiring . Then our Haar measure on is defined as by the Iwasawa decomposition . Note that . On , , , and , we always use the product measures of their factors. Let denote the characteristic function of a set . We omit its domain as we guess easily from context.
2.1. Convergence lemmas
Let be the adjoint representation of on . For , let be the representing matrix with respect to the -basis of . For , set
Then, for any and . The norm of an adele point is defined by . We remark that the norm is a -invariant and bi--invariant function on taking values in . It satisfies for all and , , where is a Siegel domain of and is defined as
Lemma 2.1.
For and , set . If , then the series converges absolutely and locally uniformly in ; moreover, the function is bounded on .
Proof.
Set . Let be a compact neighborhood of the identity of such that . Then is a disjoint union for any . Set . We have , and thus
for any and . From this,
Thus it suffices to show that the function on is integrable if . For , set If and , then by the Cartan decomposition , we have
From this, the convergence of for is evident. If and , then by the Cartan decomposition , we have
This completes the proof of the pointwise convergence. It remains to confirm the local uniformity of the convergence. Let be a compact subset of . Then, for any and . From this, is majorized term-wisely by the convergent series for . ∎
Proposition 2.2.
Let be a function such that for all and on with . Then, the series
converges absolutely and locally uniformly. Moreover, the following holds.
-
(i)
with the implied constant independent of .
-
(ii)
For any , the function belongs to for any .
Proof.
Let be a compact subset of . Then
From this, the series is dominated by , which is convergent if by Lemma 2.1.
(i) From , we have
for any . Since is bounded in , we are done.
(ii) From (i), for a fixed , the function in is bounded. Since is of finite volume, for any . ∎
2.2. Matrix coefficients of the discrete series
We recall a basic material from [14, §11, 14] to fix notation. For , let be the discrete series representation of with minimal -type , where is the character of given by . The matrix coefficient of is defined by
where is a unit vector belonging to . It satisfies the conditions :
-
(a)
with ,
-
(b)
for ,
-
(c)
.
These conditions uniquely determine the -function as
| (2.1) |
Let . We denote by the function on . Then,
| (2.2) |
Lemma 2.3.
Let . Let and a -function such that for and such that . Then we have the relation
Proof.
Set for , where . Then satisfies the same conditions (a) and (b) as . By the uniqueness of the solution to the differential equation (a), there is a constant such that for all . By putting , we have . Substituting the equation , we have
as desired. We refer to [14, Proposition 14.4] for the last identity. ∎
2.3. Green functions on over non-Archimedean local fields
Let . For such that , it is easy to show that there exists a unique function with the properties:
-
(a)
for any , .
-
(b)
.
-
(c)
, .
Here and . We denote this function by . By the Cartan decomposition, it is explicitly written as
We also have
| (2.3) |
Let be a compact set of the half plane . Then, from the formula above,
| (2.4) |
Lemma 2.4.
Let be a smooth function such that for all , and . Then we have
as long as the integral of the left-hand side is absolutely convergent.
2.4. The kernel functions
Let be a non-zero ideal and an element of such that for all . Let be a finite set. For any , set
where . Define a function on by
| (2.5) |
Lemma 2.5.
Let be a compact subset of . Then the series (2.5) converges absolutely and
with being the minimum of the set .
Let denote the space of all those functions square integrable on and for all . Then is contained in by Wallach’s criterion [31, Theorem 4.3]. It is known that the space is finite dimensional.
Proposition 2.6.
Suppose with for all and for all .
-
(1)
For any , the function belongs to the space .
-
(2)
Let be an orthonormal basis of the space endowed with the induced -inner product from . Let for each be a family such that
(2.6) for all . Then, with equals
(2.7) where is the constant (1.5).
Proof.
The assertion (1) follows from Proposition 2.2 (ii) together with the properties (a) and (b) of in § 2.2.
(2) To simplify notation, we write in place of in this proof. For any , from the well-known computation ([5]), we have
Let us write a general element as with , and . Suppose . Then , and thus . Hence the -integral yields the volume factor
The -integral and -integral are computed by Lemmas 2.3 and 2.4, respectively. To complete the proof, we use for and the relation in for which results from the self-adjointness of . ∎
Remark : Let be the cuspidal automorphic representation generated by . Then, for any . It is known to be purely imaginary by Blasius [1]. We do not need this fact in this paper.
3. Smoothed convolution product: The spectral side
3.1. Smoothed Eisenstein series
Let
be the -spherical Eisenstein series on . As a function in , it has a meromorphic continuation to , holomorphic on away from the simple pole at and satisfying the functional equation , where
We have the functional equation and the Fourier expansion
| (3.1) |
where is the global Whittaker function defined as
We have the product formula , where is the -invariant Whittaker function on determined by
| (3.2) |
if and by
| (3.3) |
if . We need the uniform estimate of the Eisenstein series:
Lemma 3.1.
For any , there exists such that, for any element of the universal enveloping algebra of , it holds that
Proof.
This follows from the materials proved in [11, §19]. Here is a brief indication of the proof. Let be a Schwartz-Bruhat function on defined as with for and for . By the multiplication from the right, acts on and hence on the space of Schwartz-Bruhat functions; by the induced action of the universal enveloping algebra of , we can define the derivative of . Let be as in the lemma. Then in [11, §19] coincides with our up to a non-zero constant multiple; this is checked by a computation on their absolute-convergence region . The desired bound follows from a similar bound of by the formula [11, (19.4)]. ∎
Let be the space of entire functions such that and such that on any interval and ,
For and , we set
Proposition 3.2.
The contour integral converges absolutely and is independent of the choice of a contour . For any ,
Proof.
Due to , the function is holomorphic on . Thus the first two assertions follow from Lemma 3.1. From Lemma 3.1, by means of [17, Lemma I.2.10], we can deduce the following estimate for the non-constant term of the Eisenstein series :
| (3.4) |
for arbitrary . To argue, we write as a sum of the following three terms:
By (3.4), the integral has the majorant on with an arbitrary large . Due to , the integrands of are holomorphic on . Thus by shifting the contour, the integrals are also shown to be bounded by on . ∎
3.2. The smoothed convolution
In this section, we fix such that for all , and consider
| (3.5) |
From Proposition 3.2 and Lemma 2.5, the integral converges absolutely.
Let be a meromorphic function on defined as
| (3.6) |
where is the constant (1.5) and
| (3.7) |
with being an orthonormal basis of . Let be an open interval. A meromorphic function on the vertical strip which is holomorphic away from possible poles on the real axis is said to be vertically of moderate growth on the strip if for any there exists such that on any . For example, Lemma 3.1 shows that the functions with are vertically of moderate growth on . Since and are finite sets, we see that the function is also vertically of moderate growth on and is holomorphic away from the possible simple poles at . By the spectral expansion given in Proposition 2.6 (2), we have
| (3.8) |
for all .
Let us calculate (3.7) when is square-free. For , the conductor of divides the ideal . Recall the basis constructed in [24] (see also [30]), where the index set consists of all the mappings such that for all and such that
where are -Whittaker functions; for , is given as
and for , and will be recalled in the proof of Lemma 3.3. By the unfolding procedure, the Rankin-Selberg integral
for the cusp forms is shown to be decomposed into the product of over all , where
| (3.9) |
is the local zeta integral. Recall the quantity defined by (1.1). For computations over a finite place , we use the decomposition
| (3.10) |
Lemma 3.3.
For any , we have
For with , we have
| (3.11) | ||||
| (3.12) |
where is the number defined in (1.1). For with , we have
| (3.13) |
Proof.
If , the integral (3.9) for is computed in the proof of [26, Lemma 6.4]. (Note that in [26] coincides with according to our notation). In the rest of the proof, let . Suppose . Then, is a unitarizable spherical representation of . Thus and with . By [30, (2.30)], we have the formula (3.11). From the formula of ,
As for the first term, by (3.10),
Since , the first integral of the right-hand side is seen to be equal to . Similarly, by the equation and by a simple variable change, each of -terms is computed to be Thus, we have
Similarly, by the relation which follows from the unitarity of and by (3.2), the integral is computed as
Hence, we obtain
In a similar fashion, the equality holds. All in all, we have the formula (3.12).
For the last formula (i.e., ), we refer to the proof of [30, Lemma 2.14]. ∎
4. An overview of the geometric side
4.1. Basic assumption
From this section on, until otherwise stated, we let be a non-zero ideal of , an element of , and a finite set of . We keep the following assumptions on :
-
(i)
The ideal is square-free.
-
(ii)
The two sets , are mutually disjoint.
-
(iii)
The weight is large in the sense that .
Let be the set of all the dyadic places in . After § 7.3, we further suppose
-
(iv)
is disjoint from .
-
(v)
The place of splits completely in .
In this paper, and are fixed. Until §8, the ideal is also fixed. To simplify notation, and are abbreviated to and , respectively.
4.2. An overview
From the classification of conjugacy classes of , we have
| (4.1) |
where the four terms on the right-hand side are defined as follows ([5], [12]).
| (4.2) | ||||
| (4.3) | ||||
| (4.4) | ||||
| (4.5) |
where is the set of all quadratic division -subalgebra and viewed as an -elliptic torus of . Then, from Lemma 2.5 and Proposition 3.2, for each type of conjugacy classes , we know that the integral
| (4.6) |
is absolutely convergent for . In the succeeding three sections, by computing this integral quite explicitly, we shall show the following:
Theorem 4.1.
For any lying on some domain , there exists a meromorphic function in on a vertical strip containing , which is holomorphic away from , is vertically of moderate growth, and satisfies
| (4.7) |
Fix with . From (3.8) and (4.1),
for all and . By applying Lemma 4.2 below, noting (3.8) and (4.1), we obtain the trace formula in the form as a meromorphic function on .
Lemma 4.2.
Let be a meromorphic function on a strip , which is vertically of moderate growth and is holomorphic away from possible simple poles at . Assume that there exists , such that for all . Then we have identically on the strip .
Proof.
Set . The function belongs to due to the assumption that is vertically of moderate growth and to the presence of the factor . For , by assumption, we see
Thus in . Since is continuous, we have the pointwise equality for all . By the holomorphicity, identically. ∎
5. The singular terms
5.1. The identity term
We shall see . Let . Since is convergent,
where and denote the integrals over the subdomains and , respectively. By writing as with , and by changing the order of integrals, we have
As for , shifting the contour to and then in the same way as above, we have
Thus, by the residue theorem, we see that equals . By , we obtain . If we set , then this shows that Theorem 4.1 is valid for the identity term.
5.2. The unipotent term
By noting , from (4.3) and (4.6), we have
Here is the constant term which is computed as
By substituting this to the formula of and by exchanging the order of integrals formally, we encounter the integral
| (5.1) |
In order to analyze this, we consider the local integrals
for any , where denotes the -th factor of .
Lemma 5.1.
-
(i)
For any and such that ,
-
(ii)
For any and such that and , we have
-
(iii)
For any and such that , we have
-
(iv)
For any and such that , we have
Proof.
(i) By (2.1) and by using [9, 3.194, 3], we obtain
where is the beta function. By using the formulas and , we are done.
(ii) From (2.3), . We compute the integral by dividing the integral domain to and to obtain the desired formula.
(iii) Let and . Then, if and only if or . Thus, is computed as the sum of the integral and the integral .
(iv) Since , we have the formula immediately. ∎
Suppose that , and are satisfied. Then from Lemma 5.1 we see that the integral (5.1) converges absolutely and have the formula
which gives a meromorphic continuation of to for a fixed . By changing the order of integrals, we obtain
where we set
with
| (5.2) |
Then we obtain Theorem 4.1 for the unipotent term with .
6. The -hyperbolic term
In this section, we study to show Theorem 4.1 for the -hyperbolic term.
6.1. Spherical functions
First, we recall the explicit formula of -invariant spherical functions on . Let and . By the Iwasawa decomposition , we have a well-defined smooth function such that
| (6.1) | |||
where
The simple poles of the factor of may cause singularities of , but they are removable due to the obvious functional equation . For any , we collect several easily proved formulas for later use:
| (6.2) | |||
| (6.3) |
Lemma 6.1.
The function is the unique complex-valued smooth function on such that which satisfies the following conditions:
| (6.4) | ||||
| (6.5) | ||||
| (6.6) |
where for is the Casimir operator of .
Proof.
Let . In the same way as [30, Lemma 5.2], for ), we deduce a recurrence relation
and from (6.5) and , which can be solved uniquely by . By the Iwasawa decomposition and (6.4), we are done. Let . From [10, Proposition 4.3] for , we have the differential equation
and from (6.6) and , which has the unique solution
By the Gauss connection formula (on the last line of [16, p.47]), it turns out that with . By [30, Lemma 3.1], we are done. ∎
6.2. Unfolding and contour shifting
From Lemma 2.5 and Proposition 3.2,
which justifies the applications of Fubini’s theorem in the following computation:
| (6.7) |
where for and , we set
| (6.8) |
which is shown to be absolutely convergent by Proposition 3.2 for all . By (3.1), the value is expressed as the sum of the following three contour integrals:
| (6.9) | ||||
| (6.10) |
where . For any , by shifting the contour to (), we have the inequality
which yields the bound for . Combining this with the bound , which is immediate from (6.9) with , we have
From this, the integral
is seen to be absolutely convergent for all . In the same way as [30, Lemma 7.6], using the residue theorem for such that , we have
In a similar manner as above, we have the estimation
for any . When we shift the contour to the left to obtain the majorant , we note that the singularity at of is canceled with the zeros of . Hence the integral
is absolutely convergent for all , and is evaluated as Consequently, . The absolute convergence of the integral
for is confirmed by the inequality
combined with the bound
which follows from (3.2) and (3.3). By Lemma 6.1, on the region , for all , we have
Lemma 6.2.
For all such that and , we have
Proof.
Let denote the integrand of the integral on the right-hand side. Suppose and . By taking so that and , Cauchy’s integral theorem yields
Hence, we have the expression of in the assertion for . We remark that the first term is holomorphic on by and , and that the second and the third terms are entire due to . ∎
By shifting the contour to () after substituting , from Lemma 6.2, we have the expression
| (6.11) |
6.3. The orbital integrals
By substituting the expression (6.11) for the last formula of (6.7) and formally changing the order of the integral and the summation, the integral
| (6.12) |
emerges naturally. In this subsection, we prove the following proposition, which is sufficient to legitimatize the formal computation explained above.
Proposition 6.3.
Suppose .
-
(1)
Let . The integral (6.12) converges absolutely for . On that region, we have the product formula with
where is the -th factor of . We have unless , where .
-
(2)
Set
For any and , there exists a positive number independent of and such that
for all and , where
For the proof, we use the following explicit formulas of the local integrals , which are proved later in §10.
Theorem 6.4.
Let and .
-
(1)
Suppose . Then, on the region , we have
with . Here is the Legendre function of the 1st kind which is defined for points outside the interval of the real axis ([16, §4.1]).
-
(2)
Let . Then, we have
In particular, if .
-
(3)
Suppose . Then, we have
-
(4)
Suppose . Then, for we have
Lemma 6.5.
We have the inequality
for all and .
Proof.
We suppose and set and . Then, by Theorem 6.4 (2), is the product of and
with . From the second expression, is seen to be holomorphic away from . Let ; then and
Thus for . From this we have by the obvious functional equation . ∎
Corollary 6.6.
Let . Given , we have a constant such that
for and , where is the ring of -integers in .
Proof.
Choose such that for all . We fix such that for all , and set . Then for all with , and for all with . Setting , we are done by Lemma 6.5. ∎
Lemma 6.7.
We have the estimate
for , and .
Proof.
Let , . Set , and
with . Then Theorem 6.4 (3) shows that is the product of and . If , we have and
Hence for . By the obvious functional equation , we also have for . Thus
∎
Lemma 6.8.
Let . On the region and ,
Proof.
Put . Suppose and let and . From Theorem 6.4 (4), a computation shows the identity with
where . Suppose . Then . From and , we have and . By these,
Thus for . Since , we then have for .
Suppose and set . Then
∎
Corollary 6.9.
On the region , , we have
for all .
Proof.
If , then . By this remark, we can deduce the inequality from those in Lemma 6.8. ∎
Lemma 6.10.
Let . On the region , the integral converges absolutely and is a holomorphic function in for any fixed . We have
uniformly for lying in a compact set of .
Proof.
Recall is given by (2.1). By the Cartan decomposition, we have
for any , and . From now on, we suppose and . We have , where
with , . We examine the convergence of these integrals separately. By the obvious estimate ,
Since the integral is convergent for all , we have the estimate compact uniformly in . In the same way, by the estimate ,
From the defining formula (6.1), we easily have the estimate as . From this the integral in the majorant of is seen to be convergent for . ∎
Corollary 6.11.
Let and . We have the estimate
uniformly for such that .
Proof.
Let be a neighborhood of in such that for all . From Lemma 6.10, we have the estimate on uniformly in . To have a bound on , we use Theorem 6.4 (1). By Stirling’s formula, is vertically bounded on . By the formula [16, line 12, p.184],
For varying and , the last integral is bounded by a constant. Thus, for and . ∎
6.4. Proof of absolute convergence
Proposition 6.12.
Suppose for all . Let . The series converges absolutely and uniformly for and locally uniformly in defining a holomorphic function of on the region
Proof.
Let be such that and for all . Then we may take in Proposition 6.3, by which the proof is reduced to showing the convergence of with being a function on the adeles defined as
Let be the ring of -integers in , i.e., . Set for and if . Let be a compact neighborhood of in such that is constant on cosets if . Then if and if . Hence if we set , then . Since is a discrete subgroup of , by choosing small enough, we have
Since for all , the last integral is easily seen to be convergent. ∎
We have
with . By the estimate of Proposition 6.12, we can change the order of the integrals to have the formula
Define
| (6.13) |
Then we obtain a precise form of Theorem 4.1 for the -hyperbolic term.
Proposition 6.13.
Suppose for all . The function is holomorphic away from and vertically of moderate growth on . For any contour , we have the formula (4.7) with .
7. The -elliptic term
In this section, we study to show Theorem 4.1 for the -elliptic term. From this section on, we suppose that splits completely in the extension ; thus for all .
7.1. Parametrization of elliptic elements
Set . Let us say that two elements and from are -equivalent if there exists such that . The -equivalence class of a pair is denoted by . The quotient set of by the -equivalence relation is denoted by , i.e.,
Let be the set of such that the polynomial is -irreducible. For , fix its representative once and for all and set and . The -conjugacy class with characteristic polynomial is represented by the element . For a quadratic extension of with a prescribed square root , let be the -algebra embedding
Then the centralizer of in is .
For any place , we can write the image of in as with ; we suppose (a) , belongs to , or (b) and . We fix such a factorization of in . Since is assumed to be completely split in , we have thus may suppose for all . Let denote the relative discriminant of . From this, it is easily seen that if or , , and for , . Let be idele class character associated with the quadratic extension by class field theory. Then and the conductor of is .
We have the direct sum decomposition as -vector spaces, which determines an -embedding as
Set . For , we set .
Lemma 7.1.
Let . If , then for all .
Proof.
If , then and . Since , this is impossible by Hensel’s lemma. ∎
Lemma 7.2.
Let .
-
(i)
If and , then .
-
(ii)
If and , then .
-
(iii)
If and , then .
-
(iv)
If and , then
-
(v)
If , , then .
Proof.
(i) For , we have . Then, if . If , by putting and with and , we have , and by Lemmma 7.1. Thus we are done. (ii) and (iii) follow from the observation that an element belongs to if and only if . (iv) Set for . Then with is written as By , we are done. (v) is confirmed by direct computation. ∎
For , set
Then we have the relation
| (7.1) |
Since for almost all , the system belongs to . Define
If we view as a closed subgroup of , then
| (7.2) |
For each place , we fix a Haar measure on by
and transfer this to by . We transfer the Haar measure on to to define a Haar measure on . We use the relations (7.1) and (7.2) to define Haar measures on groups and .
Lemma 7.3.
If , then unless , , in which case If , then .
Proof.
Let . Then, with , and , as in the proof of [14, Lemma 7.39]. We have
This equals unless and . If and , then and ; thus with . Since , we have . Hence as desired. Let and . Then is a direct product of and and , where is the Lebesgue measure on and . Thus . ∎
Using the relation (7.2), we compute
| (7.3) |
where denotes the element of such that if and if , and
| (7.4) |
7.2. Periods of Eisenstein series along elliptic tori
We shall calculate the integral (7.4), which is absolutely convergent due to the compactness of . To attain this, let us recall the multiplicity one property of the Waldspurger model of the principal series and the explicit formula of associated spherical function.
Lemma 7.4.
Suppose . Let be a -invariant vector determined by , and set
Then we have
Lemma 7.5.
Let . Then there exists a unique smooth function such that with the properties:
-
(i)
for all and .
-
(ii)
if .
-
(ii)’
if .
Proof.
By exchanging the order of integrals,
| (7.5) |
with
From Lemma 7.5, this is decomposed into a product as
| (7.6) |
Let and set . Since , by noting , we have
| (7.7) |
Lemma 7.6.
Let and . Then
where we set for and for .
Proof.
Let us recall the formula
where if and if . Noting , we have
with
where and .
(i) Suppose and . Then we have
by the variable change , and by the relation .
(ii) Suppose . Then we easily have
with and . From ,
by making the variable change and noting .
(iii) Let . By the same computation as in (ii), we have
We decompose the set into the disjoint union of , and and write , where with . We have
and easily. Since , we have that is a square if and only if is a square in . In particular, for . By this remark, we compute
where is the number of all such that . Hence
by using , , and . We omit the detail of the computation for archimedean cases (iii) and (iv), which are elementary.∎
The following formula is originally due to Hecke ([20, Chap. II, §3]).
Proposition 7.7.
We have
where and denotes the relative discriminant of .
Proof.
Lemma 7.8.
For and ,
uniformly in and such that , where .
Proof.
The function is holomorphic and vertically bounded on . By two estimates for and for , a well-known argument by the Phragmen-Lindelöf principle yields a bound uniformly valid for and for any non-square . From this and Proposition 7.7, we have the desired bound easily. ∎
7.3. Explicit formulas of local orbital integrals
For , let be the -component of the test function on . For , set
| (7.9) |
with if and if .
Theorem 7.9.
-
(1)
Suppose . Then for , we have
-
(2)
Suppose or with . Then we have
if , and otherwise
-
(3)
Suppose with . Then
-
(4)
Suppose . Then
-
(5)
Suppose . Then for , we have
Proof.
The case will be treated in § 10. Suppose . Since , we have , which in turn implies . Recall . Set and . Then, by adjusting the measures, we have
where is the integral treated in § 6.3. The case of (1) follows from Theorem 6.4 (1). The first case of (2) follows from Theorem 6.4 (2) if one notes the relation by the identity for with , which is seen to be valid as . The first case of (4) follows from Theorem 6.4 (3) similarly. The case in (5) follows from Theorem 6.4 (4) by if and by if . ∎
Lemma 7.10.
Let . Then
If , then . If , then
except when and is dyadic. If , then .
Proof.
This follows from the relation . ∎
For and , we set .
Lemma 7.11.
Let .
-
(1)
Suppose , . Then
-
(2)
Suppose . Then never happens. We have
-
(3)
Suppose . Then
-
(4)
Suppose and . Then
Suppose and . Then the above equivalences hold if we replace the in the second equivalence with .
Proof.
Since , we have ; thus implies . We prove the case (1). From , we have and . Assume . Then by Lemma 7.1. Hence , which yields . If , then gives us , or equivalently , which in turns implies as . The condition implies . This completes the first two cases of (1).
Assume . Then , which yields . From this remark, we obtain the third case of (1) from the first two cases.
The remaining assertions are proved similarly. ∎
The following proposition combined with Theorem 7.9 gives us a convenient description of the local orbital integrals in terms of .
Proposition 7.12.
Let and . Write with , . Then
-
(1)
Suppose and . If , then and
If and , then and
If and , then and
-
(2)
If . Then , and
-
(3)
If . Then
7.4. The absolute convergence of the -elliptic term
In this subsection, we prove the following.
Theorem 7.13.
For any sufficiently small , we have
| (7.10) |
uniformly in such that , .
To prove this, we need estimates of local orbital integrals, which are given by the following lemmas.
Lemma 7.14.
-
(1)
We have if , and
-
(2)
We have if , and
-
(3)
Suppose and . Then for any , we have
for any with .
-
(4)
Let . For any ,
uniformly for on the strip .
-
(5)
Let . For any subinterval , we have
uniformly for on the strip .
Proof.
When , the estimates in (1), (2) and (3) follow from the proof of Lemma 6.5, Lemma 6.7 and Corollary 6.9, respectively. The proofs are modified to be applied to the case . The estimate (4) is given by Corollary 6.11. From §10.2.1, to prove (5) it suffices to estimate , which is majorized by the sum of
∎
To be precise, let denote the natural embedding . As usual, and for will be abbreviated to and , respectively.
Lemma 7.15.
Let be such that and
-
(a)
For all , ,
-
(b)
.
Then is relatively prime to . If we define a divisor of by and set , and , then , and .
If further satisfies the condition
-
(c)
is not a square residue for some ,
then .
Proof.
Let . Then . From (a), and are relatively prime in . Hence . From the relation with , we conclude and . If , from Theorem 7.9 (3) and (1.3), the non-vanishing of implies . From Lemma 7.12 (1), we have , which causes a contradiction when combined with and . Hence we must have and as desired. Let . Then . Hence and . From Theorem 7.9, Lemma 7.11 and Proposition 7.12, the non-vanishing of implies that if and if . Therefore, and for all as desired. Suppose . Then is prime to as shown above. Hence for all , we have and . From (c), there is such that . Hence from Theorem 7.9 (4), we obtain a contradiction. This shows . ∎
Let be the set of such that and for all with an idele without -component.
Lemma 7.16.
Let be our test function on (which depends , and ). Let be an element whose minimal polynomial is with . If with , then .
Proof.
Let . Then . Hence, from , we see that there exist , and such that . Hence and . Since and and since , for almost all ’s, we have and . ∎
Let be the class number of and a complete set of representatives of the ideal class group of such that are prime ideals different from . For , set and
Let be the set of indices such that . For each , by fixing an element once and for all, we obtain a bijection by sending to . By Dirichlet’s unit theorem, the quotient group is finite.
In the following lemma, we set for convention.
Lemma 7.17.
For any , there exist , , and such that and for all .
Proof.
To argue, we fix a representative of . From definition, there exists a finite idele with such that and for all . Choose such that the ideal is or for all . Let be a fractional ideal of defined by the idele , i.e., . There exists a unique and an element such that . We set and . Then , and with some such that for all . We further write with , , and set . Then and with for all as desired. ∎
For and a place , by as before, we set , which is an element of determined only up to proportionality constants from . We remark that for unless from Lemma 7.11 and Theorem 7.9. For the arguments below to work, we need to make the following assumption (7.11) so that any representative as in Lemma 7.17 satisfies the condition (a) in Lemma 7.15:
| (7.11) | The ideal is relatively prime to all of the ideals . |
We fix prime ideals satisfying (7.11) for invoking Chebotarev’s density theorem. From Lemmas 7.16 and 7.17, we have a majorant of the series (7.10) by replacing the summation range with the set of all those pairs with , with , and . Let be a small number. In the following all estimations are uniform in such that . By the evaluations of local orbital integrals obtained in this section and by Lemmas 7.8 and 7.15, we see that (7.10) is majorized by the sum of over all tuples of an integral ideal dividing , an integral ideal dividing , , and an element which is of the form with , and , where is defined to be
with being the set of all such that , , and such that , . Here and we remark that depends on through the relation . For as above, the subset is defined as follows: if is not a square residue for some , and otherwise. This comes from Lemma 7.15.
From Lemmas 7.10 and 7.11, if , , , then and . In a way similar to Corollary 6.6, from Lemma 7.14, we have
with a constant , uniformly in and as above.
Suppose for a while. Applying Lemma 7.14, we see that is majorized by the product of
where we set for . Since (see the last part of the proof of Proposition 7.7), it is easy to confirm the identity
by the product formula and by the relation for . Due to this, the above quantity is further majorized by
Let us examine the -factor. If , then and . Hence . If , then . When , we have and is a square residue modulo . Since is non-dyadic, this implies is a square in , or equivalently and . When , we have by . Thus , and hence . Thus we have if . Therefore, ,
Here, by noting , the factor is bounded by
By this and by the majorization , we have that is majorized by the product of
and
| (7.12) | |||
By the embedding , any fractional ideal of is viewed as a -lattice of -dimensional real vector space . Let us examine the condition . Set . We use the Euclidean norm on for estimations.
Lemma 7.18.
For any with , and a non-zero ideal such that , it holds that
Proof.
From , there is an integral ideal such that . By taking the norm, we have on one hand. On the other hand, by the geometric-arithmetic mean inequality,
Thus Set . Then from the inequality ,
∎
Set
for . Set . From Lemma 7.18, the series (7.12) is bounded by
| (7.13) |
where denotes the -lattice in . The function is continuous on satisfying the majorization .
Lemma 7.19.
Let and define a function on as . Suppose . Then we have a constant such that for any and for any pair of -lattices of full rank in , we have
with , and .
Proof.
This follows from the proof of [27, Theorem A.1]. ∎
Set . Since are all contained in , from Lemma 7.19, the series (7.13) is absolutely convergent if and is small enough, and majorized by
with and with . By a slight modification of the proof of [27, Lemma A.7], we have and . From Minkowski’s convex body theorem, we have an upper-bound . Hence
As a consequence, we obtain a majorant of the series (7.13) as
From the argument so far, we obtain the half of the following majorization when .
Lemma 7.20.
Let be small enough. Let be a square-free integral ideal satisfying the assumption (7.11), a divisor of , , and . Then there exists a constant independent of such that the following inequality holds uniformly in with , , where and :
Proof.
The case is settled before the lemma. Suppose . In this case we have to estimate the extra terms with in . Set . By , (from Lemma 7.17) we have . Combining this with the relation , we have for all . Thus the -factor is bounded by by Lemma 7.14 (3). The equality means that is a square residue at all . Hence for all . Thus from the case of in Lemma 7.14 (2), we see The archimedean component to be accounted for is the product of , which is bounded on the vertical strip . The conditions on and come from and Lemma 7.14 (3), (5). ∎
7.5. The conclusion
Substituting this to (7.3) and exchanging the order of integrals, we obtain
We need to legitimatize the order exchange of integral above. By Fubini’s theorem and Lemma 7.8, it suffices to show the following.
Lemma 7.21.
For a fixed , there exist and a finite set containing such that
| (7.14) |
uniformly for all finite sets containing and for all in the strip with small .
Proof.
For simplicity we argue assuming . (The case is easier.) Let be the union of and the set of such that does not hold. If is such that , then , satisfies and . Thus as we have seen in § 7.3,
If is such that , then since , from the proof given in § 10.2.2, we also have . Thus the left-hand side of (7.14) is independent of containing . At places in , the absolute convergence of the integral and their necessary polynomial bound are shown or are easily deducible from the arguments in § 6.3 and in § 10. ∎
Set
| (7.15) |
Now we obtain Theorem 4.1 for -elliptic terms in the following form.
8. Proof of the main theorems
For any holomorphic function on and for any , the multi-dimensional contour integral is defined as the iteration of the one-dimensional contour integrals in any ordering .
In this section, we let the square-free ideal vary in such a way that is prime to , where are the ideals fixed in § 7 and we set .
8.1. Proof of Theorem 1.1 and Corollary 1.2
For , we define by (5.2) for and for . Set . Recall the function defined by (3.6) and the explicit formulas of stated in §5, §6 and §7. From §4, we have the identity
for and with (cf. Proposition 6.13 and Theorem 7.22). To obtain the formula of in Theorem 1.1, we examine local conditions on posed by various -factors in Theorem 7.9. From Lemma 7.10, when we have that if and only if ; hence implies . Thus the -symbol in the first formula of Theorem 7.9 (2) is simplified to . By a similar argument, the local conditions posed by -symbols in Theorem 7.9 are reduced to the following:
-
•
If , then ,
-
•
If , , then ,
-
•
If , or , , then .
We remark that coincides with unless or , in which case it equals . By this, we can write the condition above as follows :
-
•
and is unramified and non-trivial, then ,
-
•
or , then .
The second condition follows from . Indeed, from the relations , and , we see . Hence . This completes the proof of Theorem 1.1.
Take any . For , set
with , . We note that the ideal is implicit in the definition of . Set
for as above. By taking a multi-dimensional contour integral, we obtain the identity
| (8.1) |
from Theorem 1.1. By (3.6) and the formula
we see that the left-hand side of (8.1) coincides with (1.6) multiplied by . This completes the proof of Corollary 1.2.
8.2. Proof of Theorem 1.3
Let . Then the factor of is bounded in absolute value by from below. The value of at is of the form
with some quantities independent of . This is bounded from below by . Hence to prove Theorem 1.3, it suffices to show the hyperbolic and the elliptic terms are bounded from above by with some uniformly on the strip .
Let and a complex variable. Let be the space of all Laurent polynomials in which is invariant by the substitution . Set for . Then as is well known, the elements form a -basis of the space . For , let be the -linear span of functions . For an integral ideal with the prime decomposition with , set .
Lemma 8.1.
Let . We have for all if .
Proof.
Let us take a test function .
Lemma 8.2.
Let . Suppose satisfies . Set if and otherwise. Then , and for any small ,
| (8.2) |
uniformly for on the strip .
Proof.
From (6.13) and Theorem 6.4 (2) and (3),
| (8.3) | ||||
| (8.4) |
where and denotes the -integer ring of , and we set for . By Theorem 6.4 (4) and Lemma 8.1, the -factor (8.4) vanishes unless for all . Hence we may suppose that the summation in (8.3) is over the set . From Corollary 6.9, the -factor in (8.4) for is majorized by uniformly in . Hence as in the proof of Proposition 6.12, we have that for any small ,
uniformly on with , where if and if . The sum is majorized by
Let denote the majorant. Since are -lattices of full rank in , we apply Lemma 7.19 to estimate the sum by uniformly in the ideals and . Therefore,
Suppose . Then we can choose so that . Thus,
Suppose . Then . Thus,
This completes the proof. ∎
Lemma 8.3.
Set if and if . Then , and for any small ,
uniformly in on the strip .
Proof.
In this proof, we set . From (7.15) and Proposition 7.7, by changing the order of the integral and the summation,
| (8.5) | ||||
| (8.6) |
Here is the dyadic factor of the formula of in Proposition 7.7, and equals or according to “ or ” or not, respectively. We choose the representatives of as in Lemma 7.17. For such that , we have from ; since is non-dyadic and is a square residue modulo , we have by Lemma 7.1. Then Proposition 7.12 implies . From Theorem 7.9 and Lemma 8.1,
| (8.7) |
unless for all . Thus we may suppose for all such that . Combined with the constraint , this implies the vanishing of the -factor (8.6) except for finitely many . Thus is majorized by the sum of all with for all . Then we invoke the estimate Lemma 7.20 for each of these to obtain the majorization uniformly in with and
where and . We divide the sum to two parts and according as or not. Set . Then is over those pairs of ideals such that , and . As such,
Since is non-positive,
where we use the estimate .
The sum is over all pairs of ideals such that , , . By and by noting , we have trivially
Thus
Suppose . Then for sufficiently small , we have for . Then
Suppose . Then for . Thus
Note for . This completes the proof. ∎
Remark : With a bit more work, it can be shown that the implied constants in Lemmas 8.2 and 8.3 are taken to be of the form with being a norm on the finite dimensional space .
Theorem 1.3 (1) for follows from Corollary 1.2 combined with Lemmas 8.2 and 8.3 immediately. The assertion of Theorem 1.3 (1) for is proved by a standard argement (cf. [19, Proposition 2]), where the non-negativity of the measure is indispensable. Indeed, for any and any , the Weierstrass approximation theorem allows us to take a polynomial function such that . We set and for simplicity. By the condition (P), we have . Furthermore, there exists such that and for all with . As a result, we have
This completes the proof as varies in the compact set .
As for the proof of Theorem 1.3 (2), we first show that, under the assumption (P), the limit formula in Theorem 1.3 (1) is valid even for with , which is discontinuous. We set and for simplicity. Put for any . Then is monotonously increasing for each . For any fixed , take such that , and such that . By Theorem 1.3 (1), there exists such that both and hold for all with and for all . Then, for with is majorized by
This completes the required convergence for .
Let us return to the proof of Theorem 1.3 (2). Let be the set of all the prime ideals relatively prime to . Set . As shown above, holds with the convergence being uniform in . Let us define by the same formula as reducing the summation range from to . By the uniform bound ), we easiy confirm the difference tends to as uniformly in . Hence there exists such that for all and for all with . Hence for any with and any , there exists such that . We have since is prime. This completes the proof. We remark that in the proof above the condition (P) is not necessary when , in which case belongs to .
9. The cuspidal case
Let be the automorphic kernel function constructed in § 2.4. Let be an irreducible cuspidal automorphic representation of with trivial central character which is everywhere unramified, i.e., for all . Thus we have a set of numbers and such that for all . Let be the new vector of . Since is rapidly decreasing on the Siegel set of , the integral
| (9.1) |
converges absolutely. By replacing with in every occurrence, almost all proofs to compute (3.5) so far also work for the integral (9.1) by a slight modification. We end up with a formula resemble to the one in Corollary 1.2. To describe it, we need further notation. For , we consider the sum of triple product of cusp forms
as (3.7). We modify the definition of in § 1.2 by replacing the parameter with in the -factor for all . i.e.,
For any quadratic field extension with a prescribed square root of , we define
Theorem 9.1.
Proof.
In accordance with (4.1), the integral (9.1) breaks up to the sum of four terms with . By the cuspidality of , it is seen easily that . By the same argument as in § 6, the hyperbolic term becomes
with being the Hecke’s integral, which is identified with the central value of -function: . Since has no constant term in the Fourier expansion, the first part of § 6.2 is irrelevant; the absolute convergence is shown as in § 6.4. In the same way as in § 7, the elliptic term becomes
Since (cf. [30, Lemma 2.13]), from Lemma 7.8, we obtain the majorization , which should be a substitute of Lemma 7.8. By Theorem 7.9 and the arguments in § 7.4, we have the absolute convergence and the desired formula. ∎
10. Explicit formulas of local orbital integrals
In this section, we prove Theorems 6.4 and 7.9 by separating cases as in the following table.
To ease notation, in the archimedean cases (§10.1.1 and 10.2.1), we write and for and , respectively. In the non-archimedean cases (§10.1.2, §10.1.3, §10.1.4, §10.2.2, §10.2.3 and §10.2.4), we omit the subscript of , , , and and write them , , , and , respectively.
10.1. Local hyperbolic orbital integrals
Let . In this subsection, we compute the hyperbolic local orbital integral
where is the -th factor of .
10.1.1. The proof of Theorem 6.4 (1)
Suppose , so that . Fix .
Lemma 10.1.
On the region , we have
where for is defined as
| (10.1) | |||
Proof.
Set
By the Cartan decomposition, we have
for any , and . Since as and as , the integral converges absolutely for and defines a holomorphic function. From now on, we suppose and . Then,
which is valid for . For any ,
by the formula in the last line of [16, p.85]. From now on suppose . By this, we see that for is the product of
| (10.2) |
and
| (10.3) |
By , the formula (10.3) becomes
To have the last equality, we made the variable change . The -integral is computed as
in terms of the parabolic cylinder function by the first formula on [16, p.328]. Hence,
by the formula on [16, p.287]. Applying the integral expression
obtained from the formula on the last line of [16, p.274] by an obvious variable change, and then using the formula
which is deduced from the first formula of [16, §7.5.2] by applying ([16, p.304]), it turns out that is the product of
and
where . We note that the integral is convergent for . We apply the formula to obtain the desired formula (10.1) for . Then the equality (10.1) is extended to due to the holomorphy. ∎
Let us return to prove Theorem 6.4 (1). Suppose and set ; we note if . By the Gauss connection formula for (on the last line of [16, p.47]), the integral
becomes the sum of
| (10.4) |
and
| (10.5) |
We see that these are computed as
and
Combining these evaluations with the formula of in Lemma 10.1, we obtain
using the formula , and the duplication formula . Pluging this and to , after a simple computation, we have the formula
where for , and , we set
| (10.6) |
From [16, line 14, p.153] and [16, line 18, p.164],
and
where and are the associated Legendre functions of the first kind and of the second kind, respectively. Thus we obtain the desired formula for . By analytic continuation, it remains valid on . This completes the proof of Theorem 6.4 (1).
10.1.2. The proof of Theorem 6.4 (2)
Let . Recall . For and , we have for some if and only if and . Thus, unless , in which case , where . For , we easily have that is absolutely convergent for all and
if . Hence
By (6.2), we are done. The second claim is obvious from the formula of .
10.1.3. The proof of Theorem 6.4 (3)
Let . From (3.10), we have the equality
from which we see that is times the sum of the following integrals
| (10.7) | |||
| (10.8) |
where . Since if and only if , , from the proof of Theorem 6.4 (2) in §10.1.2, the integral (10.7) equals . The integral (10.8) is computed as , where . Let . Suppose . Then we easily have , and (10.8) becomes . Hence Suppose . Noting , we have
where is the same integral as in the proof of Theorem 6.4 (2) in §10.1.2. Thus (10.8) is the sum of , which becomes as before and Hence equals the following expression multiplied by :
From this we get the desired formula by a short computation.
10.1.4. The proof of Theorem 6.4 (4)
Let . Put . By applying (2.3) and (6.1) and then changing the order of integrals,
| (10.9) | ||||
where we set
As will be shown below, the integral is absolutely convergent if . Suppose . Then . Hence
if . By a computation, we obtain
From this, combined with (6.3), we get the formula
| (10.10) |
Suppose . Then and
if . Suppose . Then and
if . Hence for . From this, by a direct computation, we get the formula (10.10) again.
10.2. Local elliptic orbital integrals
Let . In this subsection, we compute the integral (7.9) with to complete the proof of Theorem 7.9. In this section throughout, we fix with the decomposition at a place as before. Set and to simplify notation. Recall the construction at the begining of § 7.2. By multiplying , the integral is transformed to
| (10.11) | ||||
10.2.1. The proof of Theorem 7.9 (1)
Let and . By and by (b) in § 2.2, we have
with
By the formula ( easily proved by the residue theorem,
Since is even, by decomposing the -integral over into positive and negative reals, we obtain
where is the Zagier’s function defined in [35, p.110]. We have
for by means of the formula [9, p.961, 8.713, 3], where the square root of is chosen so that . Consequently, by noting , we have
We complete the proof by Lemma 7.4.
10.2.2. The proof of Theorem 7.9 (2) and (3)
Since , the integral domain of (10.11) is restricted to only those such that
| (10.12) | |||
| (10.13) | |||
| (10.14) |
with some .
(I) The case .
(i) Suppose , . From Lemma 7.1, . Hence from (10.14) we get , which, combined with (10.13), yields . From the last relation in (10.12), we have the containment from which follows. If were non-unit, then , which is impossible due to Lemma 7.1. Hence we have that the existence of satisfying (10.12), (10.13) and (10.14) is equivalent to and . Thus by Lemm 7.4, we have .
(ii) Suppose , . Let us consider the case . If , then (10.14) implies . Combining this with (10.12) and (10.13), we obtain and . Hence by Lemma 7.4,
If , the condition (10.14) implies . Then using (10.12) and (10.13), we get and . Thus if and if , and whence
We note from Lemma 7.11 (4) and Proposition 7.12 (1). Next consider the case . If , then , which contradicts to (10.14). Thus . If , then (10.14) implies . We may set . Then, and . Thus, if and if . We have
(iii) Suppose . Then, implies , where is an odd power of and (10.14) is never attained. Thus .
(II) The case .
Since , the condition (10.14) yields . Thus the condition (10.12) is equivalent to , , and the condition (10.13) to . Hence,
| (10.15) |
where we put . For , we set
Lemma 10.2.
Let . If and , we have
If and ,
If , we have . If and , we have
Proof.
In the proof, we write for for simplicity. First suppose . Then is equivalent to and . Suppose . Due to Lemma 7.1, and ; thus
Suppose and . Then since , we have for . Hence is empty unless , in which case . Similarly, is empty unless , in which case it is . We also have Thus
which is simplified to the desired form. The case with is similar.
Next suppose . In the case, for all . Hence whose volume is , and whose volume is . By , we are done. ∎
Lemma 10.3.
For any such that and such that , we have
with .
Proof.
A direct computation. ∎
First we consider the case where . If is non-dyadic, from (10.15), Lemmas 10.2 and 10.3, we get
To complete the proof, it suffices to use Lemma 7.4 and to note that equals or according to or , which follows from Lemma 7.11 (1) and Proposition 7.12 (1). If is dyadic and , in the same way, we have
To complete the proof, it suffices to use Lemma 7.4 and to note that if from Lemma 7.11 (4) and Proposition 7.12 (1). The remaining cases are similar.
10.2.3. The proof of Theorem 7.9 (4)
Lemma 10.4.
Suppose . Then, we have .
Proof.
Suppose . Then . We have that if and only if there exists with the properties:
| (10.17) |
where we set . By , then , and hence the last condition of (10.17) yields . From the first and the second ones, we then have and . The third condition is impossible by Lemma 7.1. Thus for all . In the same way, for all . Suppose . Then, , and if and only if (10.17); from the last of (10.17), , which is impossible due to . Hence for all . Similarly, we obtain for all . ∎
Next we consider the case . For the computation of and , we set .
Lemma 10.5.
If ,
Proof.
The integrand of is non-zero if and only if there exists such that , , , , . By , is a unit and whence . From this, .
The integrand of is non-zero if and only if , , , , . By , is a unit and whence
∎
10.2.4. The proof of Theorem 7.9 (5)
Let and the Green function on defined in §2.3. First we consider the case . The case is treated after Lemma 10.12. From (2.3), the integral (10.11) is written as
where . By the variable change , , we have
| (10.18) |
with
When is viewed as a constant, we let denote the integrand of . First we consider the case when is a non-unit.
Lemma 10.6.
Suppose , and set . Let .
-
(i)
When ,
-
(ii)
When ,
Proof.
Suppose . Then and . Hence . By dividing the integration domain into and into , we write the integral as the sum with . Let . Then by , we have . Hence , and which implies . Hence . Thus for all and
| (10.19) |
(i) Suppose . Then and . For , we have and . Hence . For , we have and . Hence . Thus . Therefore,
We have
(ii) Suppose . Then from , we have . Thus . Hence . Moreover, . If , we have and ; thus . Hence
Since , from (10.19) we have
Therefore, we obtain the desired formula by . ∎
Lemma 10.7.
Suppose , and set . Let .
-
(i)
When ,
-
(ii)
When ,
-
(iii)
When ,
Proof.
Note that . As in the proof of Lemma 10.6, we write , with and . For , we have and , by which . Thus for . We have for .
(i) Suppose . For , we have and ; thus . The set is empty due to . Hence
For , we have and . For , we have ; thus . For , we have , which implies ; thus . Therefore,
From this,
(ii) Suppose . Then . For , we have and ; hence . For , and ; thus . Hence
We have . For , we have which in turn yields and . Hence for . We have
Here to have the second equality, we split the integral over to those over and over .
From the evaluations of and , by we are done. ∎
The following is given by a direct computation.
Lemma 10.8.
If , then
Lemma 10.9.
Let . Set . Suppose . For ,
For ,
We obtain the formula for in Theorem 7.9 (5) for the case from (10.18) applying Lemma 10.9 combined with Lemma 7.4, Lemma 7.11 (2) and Proposition 7.12 (2). We note that .
Next we consider the case when is a non-square unit.
Lemma 10.10.
Suppose and . Set . Suppose .
-
(i)
When ,
-
(ii)
When ,
-
(iii)
When ,
Proof.
Let and , and set for to write as the sum , where
(i) Suppose . Then . Since , the set is empty. If , then and by Lemma 7.1; hence . Thus
For , we have ; hence and . Thus equals
(ii) Suppose . Then . If , then , ; hence . If , then and as above; hence . Thus
For all , we have . If , we have ; thus . If , we have ; hence . Note that is empty unless . If , then and ; hence . Thus,
From the evaluations of and , we are done. ∎
Lemma 10.11.
Suppose and . Set . If ,
Proof.
Let and be as in the proof of Lemma 10.10. For , we have , , and . Hence . Thus
(i) Suppose . Then . If , then and because is not congruent to a square modulo by assumption (Lemma 7.1); hence . Thus
(ii) Suppose . Then and . If , then we have and ; hence . Thus,
Having evaluations of and , we are done. ∎
Lemma 10.12.
Let . Suppose . For ,
For ,
We obtain the formulafor in Theorem 7.9 (5) for the case from (10.18) applying Lemma 10.12 combined with Lemma 7.4, Lemma 7.11 (1) and Proposition 7.12 (1).
The case is included in the case . Indeed, we can evaluate when as
where we set and we write for to make dependence on explicit. The function is independent of by Lemma 10.6 (i) and Lemma 10.11 (i). Since is continuous at by its integral representation, we obtain the formula of by . Hence the formula in Theorem 7.9 (5) is valid when .
Acknowledgements
The second author was supported by Grant-in-Aid for Scientific research (C) 15K04795.
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Shingo SUGIYAMA
Department of Mathematics, College of Science and Technology, Nihon University, Suruga-Dai, Kanda, Chiyoda, Tokyo 101-8308, Japan
E-mail : s-sugiyama@math.cst.nihon-u.ac.jp
Masao TSUZUKI
Department of Science and Technology, Sophia University, Kioi-cho 7-1 Chiyoda-ku Tokyo, 102-8554, Japan
E-mail : m-tsuduk@sophia.ac.jp